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Mirrors > Home > ILE Home > Th. List > cnmet | GIF version |
Description: The absolute value metric determines a metric space on the complex numbers. This theorem provides a link between complex numbers and metrics spaces, making metric space theorems available for use with complex numbers. (Contributed by FL, 9-Oct-2006.) |
Ref | Expression |
---|---|
cnmet | ⊢ (abs ∘ − ) ∈ (Met‘ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnex 7913 | . 2 ⊢ ℂ ∈ V | |
2 | absf 11090 | . . 3 ⊢ abs:ℂ⟶ℝ | |
3 | subf 8136 | . . 3 ⊢ − :(ℂ × ℂ)⟶ℂ | |
4 | fco 5376 | . . 3 ⊢ ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ) | |
5 | 2, 3, 4 | mp2an 426 | . 2 ⊢ (abs ∘ − ):(ℂ × ℂ)⟶ℝ |
6 | subcl 8133 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 − 𝑦) ∈ ℂ) | |
7 | 6 | abs00ad 11045 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((abs‘(𝑥 − 𝑦)) = 0 ↔ (𝑥 − 𝑦) = 0)) |
8 | eqid 2177 | . . . . . 6 ⊢ (abs ∘ − ) = (abs ∘ − ) | |
9 | 8 | cnmetdval 13662 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥(abs ∘ − )𝑦) = (abs‘(𝑥 − 𝑦))) |
10 | 9 | eqcomd 2183 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (abs‘(𝑥 − 𝑦)) = (𝑥(abs ∘ − )𝑦)) |
11 | 10 | eqeq1d 2186 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((abs‘(𝑥 − 𝑦)) = 0 ↔ (𝑥(abs ∘ − )𝑦) = 0)) |
12 | subeq0 8160 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑥 − 𝑦) = 0 ↔ 𝑥 = 𝑦)) | |
13 | 7, 11, 12 | 3bitr3d 218 | . 2 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑥(abs ∘ − )𝑦) = 0 ↔ 𝑥 = 𝑦)) |
14 | abs3dif 11085 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (abs‘(𝑥 − 𝑦)) ≤ ((abs‘(𝑥 − 𝑧)) + (abs‘(𝑧 − 𝑦)))) | |
15 | abssub 11081 | . . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (abs‘(𝑥 − 𝑧)) = (abs‘(𝑧 − 𝑥))) | |
16 | 15 | oveq1d 5883 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((abs‘(𝑥 − 𝑧)) + (abs‘(𝑧 − 𝑦))) = ((abs‘(𝑧 − 𝑥)) + (abs‘(𝑧 − 𝑦)))) |
17 | 16 | 3adant2 1016 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((abs‘(𝑥 − 𝑧)) + (abs‘(𝑧 − 𝑦))) = ((abs‘(𝑧 − 𝑥)) + (abs‘(𝑧 − 𝑦)))) |
18 | 14, 17 | breqtrd 4026 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (abs‘(𝑥 − 𝑦)) ≤ ((abs‘(𝑧 − 𝑥)) + (abs‘(𝑧 − 𝑦)))) |
19 | 9 | 3adant3 1017 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥(abs ∘ − )𝑦) = (abs‘(𝑥 − 𝑦))) |
20 | 8 | cnmetdval 13662 | . . . . . 6 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑧(abs ∘ − )𝑥) = (abs‘(𝑧 − 𝑥))) |
21 | 20 | 3adant3 1017 | . . . . 5 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑧(abs ∘ − )𝑥) = (abs‘(𝑧 − 𝑥))) |
22 | 8 | cnmetdval 13662 | . . . . . 6 ⊢ ((𝑧 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑧(abs ∘ − )𝑦) = (abs‘(𝑧 − 𝑦))) |
23 | 22 | 3adant2 1016 | . . . . 5 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑧(abs ∘ − )𝑦) = (abs‘(𝑧 − 𝑦))) |
24 | 21, 23 | oveq12d 5886 | . . . 4 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑧(abs ∘ − )𝑥) + (𝑧(abs ∘ − )𝑦)) = ((abs‘(𝑧 − 𝑥)) + (abs‘(𝑧 − 𝑦)))) |
25 | 24 | 3coml 1210 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑧(abs ∘ − )𝑥) + (𝑧(abs ∘ − )𝑦)) = ((abs‘(𝑧 − 𝑥)) + (abs‘(𝑧 − 𝑦)))) |
26 | 18, 19, 25 | 3brtr4d 4032 | . 2 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥(abs ∘ − )𝑦) ≤ ((𝑧(abs ∘ − )𝑥) + (𝑧(abs ∘ − )𝑦))) |
27 | 1, 5, 13, 26 | ismeti 13479 | 1 ⊢ (abs ∘ − ) ∈ (Met‘ℂ) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 × cxp 4620 ∘ ccom 4626 ⟶wf 5207 ‘cfv 5211 (class class class)co 5868 ℂcc 7787 ℝcr 7788 0cc0 7789 + caddc 7792 ≤ cle 7970 − cmin 8105 abscabs 10977 Metcmet 13114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-iinf 4583 ax-cnex 7880 ax-resscn 7881 ax-1cn 7882 ax-1re 7883 ax-icn 7884 ax-addcl 7885 ax-addrcl 7886 ax-mulcl 7887 ax-mulrcl 7888 ax-addcom 7889 ax-mulcom 7890 ax-addass 7891 ax-mulass 7892 ax-distr 7893 ax-i2m1 7894 ax-0lt1 7895 ax-1rid 7896 ax-0id 7897 ax-rnegex 7898 ax-precex 7899 ax-cnre 7900 ax-pre-ltirr 7901 ax-pre-ltwlin 7902 ax-pre-lttrn 7903 ax-pre-apti 7904 ax-pre-ltadd 7905 ax-pre-mulgt0 7906 ax-pre-mulext 7907 ax-arch 7908 ax-caucvg 7909 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4289 df-po 4292 df-iso 4293 df-iord 4362 df-on 4364 df-ilim 4365 df-suc 4367 df-iom 4586 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-fv 5219 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-1st 6134 df-2nd 6135 df-recs 6299 df-frec 6385 df-map 6643 df-pnf 7971 df-mnf 7972 df-xr 7973 df-ltxr 7974 df-le 7975 df-sub 8107 df-neg 8108 df-reap 8509 df-ap 8516 df-div 8606 df-inn 8896 df-2 8954 df-3 8955 df-4 8956 df-n0 9153 df-z 9230 df-uz 9505 df-rp 9628 df-seqfrec 10419 df-exp 10493 df-cj 10822 df-re 10823 df-im 10824 df-rsqrt 10978 df-abs 10979 df-met 13122 |
This theorem is referenced by: cnxmet 13664 remet 13673 |
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